Orbital Coordinate Systems, Part I

By Dr. T.S. Kelso

September/October 1995

By this point, I hope to have helped you develop an understanding of two key
aspects of practical orbital mechanics. The first has to do with why we use the
orbital models we do for predicting the position of earth-orbiting artificial
satellites. As with any computer model, orbital models must trade off accuracy
for computational speed. Which model you decide to use will depend upon which of
these factors is most important to you.

Of course, from a practical perspective, the choice of orbital model is also
strongly influenced by the availability of data (element sets). Knowing that
orbital element sets are generated by fitting observations to a trajectory based
upon a particular orbital model is the second of our key aspects. Accuracy of
our predictions will depend upon using that same orbital model.

Up to now, however, all we've really talked about are orbital element sets.
But how do we get from the data in these orbital element sets to something we
can use, such as knowing where to look (or point an antenna) when a satellite
passes over? To answer this question requires an understanding of the various
coordinate systems involved and how to transform coordinates (typically position
and velocity) from one system to another. The correct application of these
coordinate transformations is every bit as important to our overall accuracy as
the selection of the orbital model itself.

Where do we start? Let's start with the orbital element sets themselves and
discuss some terminology. The two most common forms of orbital element sets are
state vectors and Keplerian orbital elements (e.g., the NORAD
two-line element sets). A state vector is a collection of values (states)
that if known, together with the state transformation rules (how the
state vector changes over time), the state vector for any past or future time
can be computed. For a satellite in Earth orbit, if we ignore atmospheric drag
and maneuvering, the state vector would be comprised of the satellite's position
and velocity. Knowing the position alone would not be sufficient, since a
satellite with zero velocity would fall to Earth while one with orbital velocity
would not, even if the satellites start at the same physical location.

We cannot, however, talk about position and velocity without discussing the
coordinate system that these values are measured relative to. For most state
vectors, this is the Earth-Centered Inertial (ECI) coordinate system. The
first part of this designation should seem fairly obvious. That is, since we're
studying objects that revolve around the center of the Earth, it seems natural
to have the center (origin) of our coordinate system at the center of the Earth.
Inertial, in this context, simply means that the coordinate system is not
accelerating (rotating). In other words, it is 'fixed' in space relative to the
stars. We shall see that this is an ideal definition of the ECI coordinate
system, but we won't worry about the slight rotations involved until later.

The ECI coordinate system (see Figure 1) is typically defined as a
Cartesian coordinate system, where the coordinates (position) are defined
as the distance from the origin along the three orthogonal (mutually
perpendicular) axes. The z axis runs along the Earth's rotational axis
pointing North, the x axis points in the direction of the vernal
equinox (more on this in a moment), and the y axis completes the
right-handed orthogonal system. As seen in Figure 1, the vernal equinox is an
imaginary point in space which lies along the line representing the intersection
of the Earth's equatorial plane and the plane of the Earth's orbit around the
Sun or the ecliptic. Another way of thinking of the x axis is that
it is the line segment pointing from the center of the Earth towards the center
of the Sun at the beginning of Spring, when the Sun crosses the Earth's equator
moving North. The x axis, therefore, lies in both the equatorial plane
and the ecliptic. These three axes defining the Earth-Centered Inertial
coordinate system are 'fixed' in space and do not rotate with the Earth.

Figure 1. Earth-Centered Inertial (ECI) Coordinate System

Now, while state vectors are normally used with numerical integration models
for highly accurate calculations, Keplerian orbital elements are used for the
vast majority of orbital predictions. But, the ECI coordinate system is still
often used as the common coordinate system when performing coordinate
transformations. For example, before a calculation can be made of the distance
between a satellite and an observer on the ground, both the satellite and the
observer's position must be defined in a common coordinate system. Since the
satellite's position is typically represented by a Keplerian orbital element set
and the observer's position is given in latitude, longitude, and altitude above
the Earth's surface, we cannot perform the calculation directly without first
converting to a common coordinate frame.

As it turns out, the NORAD SGP4 orbital model takes the standard two-line
orbital element set and the time and produces an ECI position and velocity for
the satellite. In particular, it puts it in an ECI frame relative to the "true
equator and mean equinox of the epoch" of the element set. This specific
distinction is necessary because the direction of the Earth's true rotation axis
(the North Pole) wanders slowly over time, as does the true direction of the
vernal equinox. Since observations of satellites are made by stations fixed to
the Earth's surface, the elements generated will be referenced relative to the
true equator. However, since the direction of vernal equinox is not tied to the
Earth's surface, but rather to the Earth's orientation in space, an
approximation must be made of its true direction. The approximation made in this
case is to account for the precession of the vernal equinox but to ignore the
nutation (nodding) of the Earth's obliquity (the angle between the
equatorial plane and the ecliptic). We'll address how to use this level of
detail in a future column.

So, we now know that whether we're using state vectors or Keplerian orbital
element sets, our calculations will likely yield ECI position and velocity.
Let's begin working now to answer two common questions in satellite tracking.
The first question is: Where do I look or point my antenna to acquire a
particular satellite? The second question is: What is the latitude, longitude,
and altitude of that satellite? These questions come up frequently, whether the
goal is to watch the US Space Shuttle and Russian Mir Space Station pass
overhead, to acquire an amateur radio satellite, or to determine the longitude
of a geostationary TVRO satellite. But, to be able to answer these questions, we
will need to determine either the position of an observer on the Earth relative
to the ECI coordinate frame or the position of a satellite relative to the
Earth. In either case, we will need to know the rotation angle between the
Greenwich Meridian (zero degrees longitude) and the vernal equinox and, hence,
the orientation of the Earth relative to the ECI coordinate frame.

Let's start by calculating the position of an observer in the ECI coordinate
frame. For our initial discussions, we'll assume a spherical Earth. This
assumption is not a particularly good one, as we'll see in our next column, but
will make the initial development easier to follow. The calculation of the
z coordinate is straightforward, as can be seen in Figure 2. This figure
shows a side cutaway of the Earth with North up. For an observer at latitude
φ, the z coordinate is shown in Figure 2, where Re
is the Earth's equatorial radius. To calculate the x and y
coordinates, we will also need the value of R from Figure 2. If we wanted
to calculate z and R for distances above mean sea level, we would
simply replace Re with Re + h, where
h is the distance above mean sea level.

Figure 2. Latitude to ECI Conversion

Computing the x and y coordinates requires a bit more work.
Since the Earth rotates in the x-y plane (i.e., about the z axis),
the x and y coordinates of a point on the Earth's surface will
vary with time, unlike the z coordinate. However, if we know the angle
between the observer's longitude and the x axis (the vernal equinox), we
can specify the x and y coordinates as a function of time. In
fact, if we designate the angle between the x axis and the observer's
longitude as θ(τ), where τ is the time of interest,
x(τ) and y(τ) are given in Figure 3. This figure shows a
slice through the Earth, parallel to the equatorial plane and through the
observer's location.

Figure 3. Longitude to ECI Conversion

Upon first inspection, these equations would seem straightforward enough. But
just what is θ(τ) and how is it calculated? The function
θ(τ) is what astronomers refer to as the local sidereal time.
Sidereal time is simply time measured relative to the stars. In our
day-to-day lives, we are used to measuring time relative to the position of the Sun
because of its obvious position in the heavens. This time scale is referred to
as mean solar time. As with any time system, time is defined as the angle
between the observer and some reference direction. With mean solar time, the
reference direction is the direction of the mean sun; with sidereal time, the
direction is the vernal equinox—just the direction we need for our
calculation. So what causes the difference between these two time scales?

As seen in Figure 4, the position of the Sun moves with respect to the stars
because of the Earth's orbit around it. Let's say we noted the position of the
Sun relative to the stars when it crosses our meridian (longitude) on one day.
By definition, that passage is called local noon. However, when that same
position relative to the stars crosses our meridian on the following day, the
Sun will not yet have reached our meridian. That is to say, the position will
cross our meridian before local noon. The interval of time between two
successive meridian crossings of a fixed position in inertial space is referred
to as one sidereal day. Sidereal midnight occurs when the vernal equinox
crosses the meridian. The interval of time between two successive meridian
crossings of the mean sun is referred to as one mean solar day. As seen in
Figure 4, the Earth must rotate a bit more for a mean solar day than for a
sidereal day. In fact, a sidereal day is only
23h56m04s.09054 of mean solar time. This
difference, while small, is extremely important.

Figure 4. Sidereal versus Solar Time

Now, since all of our common time measurements are based on UTC (Coordinated
Universal Time) which is mean solar time, how do we calculate our local sidereal
time? Well, as shown in Figure 3, the local sidereal time can be calculated by
adding the observer's east longitude, λE, to the
Greenwich sidereal time (GST), θg(τ). Oftentimes, GST (or
more specifically, Greenwich Mean Sidereal Time or GMST), can be found in
references such as the US Naval Observatory's Astronomical Almanac. If
GMST is known for 0h UTC, θg(0h), on a
particular date, then θg(Δτ) =
θg(0h) + ωe·Δτ,
where Δτ is the UTC time of interest and ωe =
7.29211510 × 10-5 radians/second is the
Earth's rotation rate. Unfortunately, this approach requires a table of
reference times to do the calculations. Another approach is to calculate
θg(0h) using the equation from Page 50 of the
Explanatory
Supplement to the Astronomical Almanac:

where Tu = du/36525 and
du is the number of days of Universal Time elapsed since JD
2451545.0 (2000 January 1, 12h UT1).

While we've covered a lot of ground in this column, we obviously still have a
bit more to go before we can answer the questions raised above. For our computer
implementation, we will first need to develop a procedure for calculating the
Julian Date in our last equation. Then, we will need to refine our conversion
from latitude, longitude, and altitude to ECI coordinates to incorporate an
oblate (flattened) Earth. When we make this refinement, we will also see the
magnitude of error which can occur if this factor is ignored. At this point, we
will have finished our first coordinate transformation and will be able to
calculate the vector from the Earth observer to the satellite. We will then
begin the process of developing our second coordinate transformation, that from
ECI to the topocentric-horizon or azimuth-elevation coordinate system. It is
this system which will allow us to measure the position of a satellite relative
to the Earth's surface.

We will also begin to include snippets of computer code to illustrate the
theory we're developing here. If you'd like to look ahead, these routines can be
found in the file sgp4-plb26a.zip
on the CelesTrak WWW site.

As always, if you have questions or comments on this column, feel free to
send me e-mail at
TS.Kelso@celestrak.com or write care of
Satellite Times. Until next time, keep looking up!